This application contains a microfiche appendix submitted on 1 microfiche sheet and 69 frames.
1. Field of the Invention
The invention is directed to an arrangement of computer elements connected to one another, methods for computer-supported determination of a dynamics that forms the basis of a dynamic process, and is also directed to a method for computer-supported training of an arrangement of computer elements connected to one another.
2. Description of the Related Art
The publication by S. Haykin, Neural Networks: A Comprehensive Foundation, discloses that an arrangement of computer elements connected to one another be utilized for determining a dynamics that forms the basis of a dynamic process.
In general, a dynamic process is usually described by a status transition description that is not visible to an observer of the dynamic process and by an output equation that describes observable quantities of the technical, dynamic process.
Such a structure is shown in FIG. 2.
A dynamic system 200 is subject to the influence of an external input quantity u of a prescribable dimension, whereby an input quantity ut at a time t is referenced ut:
utxcex51,
whereby 1 references a natural number.
The input quantity ut at a time t causes a change of the dynamic process that sequences in the dynamic system 200.
An internal condition st (st xcex5 M) of a prescribable dimension m at a time t cannot be observed by an observer of the dynamic system 200.
Dependent on the internal condition st and on the input quantity ut, a status transition of the internal condition st of the dynamic process is caused, and the status of the dynamic process switches into a successor status st+1 at a following point in time t+1.
The following is thereby valid:
st+1=f(st,ut)xe2x80x83xe2x80x83(1)
whereby f(.) references a general imaging rule.
An output quantity yt at a point in time t observable by a observer of the dynamic system 200 is dependent on the input quantity ut as well as on the internal status st.
The output quantity yt(yt xcex5 n) is a prescribable dimension n.
The dependency of the output quantity yt on the input quantity ut and on the internal status st of the dynamic process is established by the following, general rule:
yt=g(st,ut),xe2x80x83xe2x80x83(2)
whereby g(.) references a general imaging rule.
For describing the dynamic system 200, the Haykin publication utilizes an arrangement of computer elements connected to one another in the form of a neural network of neurons connected to one another. The connections between the neurons of the neural network are weighted. The weightings of the neural network are combined in a parameter vector v.
An inner status of a dynamic system that underlies a dynamic process is thus dependentxe2x80x94according to the following rulexe2x80x94on the input quantity ut and on the internal status of the preceding point in time st and on the parameter vector v:
st+1=NN(v,st,ut),xe2x80x83xe2x80x83(3)
whereby NN(.) references an imaging rule prescribed by the neural network.
The arrangement known from Haykin and referred to as Time Delay Recurrent Neural Network (TDRNN) is trained such in a training phase that a respective target quantity ytd at a real dynamic system is respectively determined for an input quantity ut. The Tupel (input quantity, identified target quantity) is referred to as training datum. A plurality of such training data form a training data set.
The TDRNN is training with the training data set. An overview of various training methods can likewise be found in Haykin.
It must be emphasized at this point that only the output quantity yt at a time t of the dynamic system 200 can be recognized. The internal system status st cannot be observed.
The following cost function E is usually minimized in the training phase:                               E          =                                    1              T                        ⁢                                          ∑                                  t                  =                  1                                T                            ⁢                                                                    (                                                                  y                        t                                            -                                              y                        t                        d                                                              )                                    2                                ⟶                                                                            xe2x80x83                                        ⁢                    min                                                        f                    ,                    g                                                                                      ,                            (        4        )            
whereby T references a plurality of points in time taken into consideration.
The publication by Ackley et al., A Learning Algorithm for Boltzmann Machines discloses what is known as a neural auto-associator (see FIG. 3).
The auto-associator 300 is composed of an input layer 301, of three covered layers 302, 303, 304 as well as of an output layer 305.
The input layer 301 as well as a first covered layer 302 and a second covered layer 303 form a unit with which a first non-linear coordinate transformation g can be implemented.
The second cover layer 303 together with a third covered layer 304 and the output layer 305 together form a second unit with which a second non-linear coordinate transformation h can be implemented.
This five-layer neural network 300 disclosed by Ackley et al. comprises the property that an input quantity xt is transformed onto an internal system status according to the first non-linear coordinate transformation g. Proceeding from the second covered layer 303, upon employment of the third covered layer 304 toward the output layer 305 upon employment of the second non-linear coordinate transformation h, the internal system status is essentially transformed back onto the input quantity xt. The goal of this known structure is the imaging of the input quantity xt in a first status space X onto the internal status st in a second status space S, whereby the dimension of the second status space Dim(S) should be smaller than the dimension of the first status space Dim(X) in order to achieve a data compression in the hidden layer of the neural network. The back-transformation into the first status space X corresponds to a decompression in this case.
The publication by H. Rehkugler et al., Neuronale Netze in der xc3x96konomie, Grundlagen und finanzwirtschaftliche Anwendungen also provides an overview of the fundamentals of neural networks and the possible applications of neural networks in the field of economics.
The known arrangements and methods particular exhibit the disadvantage that an identification or, respectively, modeling of a dynamic system that, in particular, is subject to substantial noise, i.e. whose structure is extremely complex in the time domain, is not possible.
The present invention is thus based on the problem of providing an arrangement of computer elements connected to one another with which a modeling of a dynamic system that is subject to noise is possible, the arrangement not being subject to the disadvantages of the known arrangements.
The invention is also based on the problem of providing a method for computer-supported determination of a dynamics that forms the basis of a dynamic process for dynamic processes that can be determined only within adequate precision with known methods.
The problems are solved by the arrangement as well as by the methods INSERT CLAIMS 2-26
The arrangement of computer elements connected to one another comprises the following features:
a) input computer elements to which time row values that respectively describe a status of a system at a point in time can be supplied;
b) transformation computer elements for the transformation of the time row values into a predetermined space, the transformation elements being connected to the input computer elements;
c) whereby the transformation computer elements are connected such to one another that transformed signals can be taken at the transformation computer elements, whereby at least three transformed signals relate to respectively successive points in time;
d) composite computer elements that are connected to respectively two transformation computer elements;
e) a first output computer element that is connected to the transformation computer elements, whereby an output signal can be taken at the first output computer element; and
f) a second output computer element that is connected to the composite computer elements and with whose employment a predetermined condition can be taken into consideration in a training of the arrangement.
A method for the computer-supported determination of a dynamics on which a dynamic process is based comprises the following steps:
a) the dynamic process is described by a time row with time row values in a first status space, whereby at least one first time row value describes a status of the dynamic process at a first point in time and a second time row value describes a status of the dynamic process at a second point in time;
b) the first time row value is transformed into a second status space;
c) the first time row value in the second status space is subjected to an imaging onto a second time row value in the second status space;
d) the second time row value in the second status space is transformed back into the first status space;
e) the transformation and the imaging ensue such that the dynamic process described by the time row values in the second status space satisfies a predetermined condition;
f) a dynamic of the dynamic process is determined from the time row values in the second status space.
Given a method for computer-supported determination of a dynamic that forms the basis of a dynamic process upon employment of an arrangement of computer elements connected to one another, the input signal is supplied to the arrangement and the arrangement determines a first output signal from which the dynamic is determined. The arrangement thereby comprises the following structure:
a) the dynamic process is described by a time row having time row values in a first status space, whereby at least one first time row value describes a status of the dynamic process at a first point in time and a second time row value describes a status of the dynamic process at a second point in time;
b) the first time row value is transformed into a second status space;
c) the first time row value in the second status space is subjected to an imaging onto a second time row value in the second status space;
d) the second time row value in the second status space is transformed back into the first status space;
e) the transformation and the imaging ensue such that the dynamic process described by the time row values in the second status space meets a predetermined condition;
f) the dynamic of the dynamic process is determined from the time row values in the second status space.
In a method for the computer-supported training of an arrangement of computer elements connected to one another, the arrangement is trained upon employment of predetermined training data taking a condition into consideration, whereby the arrangement comprises the following components:
a) the dynamic process is described by a time row with time row values in a first status space, whereby at least one first time row value describes a status of the dynamic process at a first point in time and a second time row value describes a status of the dynamic process at a second point in time;
b) the first time row value is transformed into a second status space;
c) the first time row value in the second status space is subjected to an imaging onto a second time row value in the second status space;
d) the second time row value in the second status space is transformed back into the first status space;
e) the transformation and the imaging ensue such that the dynamic process described by the time row values in the second status space meets a predetermined condition;
f) the dynamic of the dynamic process is determined from the time row values in the second status space.
The invention makes the modeling of a dynamic system that comprises a dynamic process possible on the basis of a smoother trajectory in a new status space, so that the further development thereof can be predicted more simply at a following point in time.
In particular, the invention achieves a better capability of distinguishing between noise and actual dynamic of the dynamic process.
In contrast to the principle of the auto-associator, a dimension reduction in the transformation into the second status space is not a criterion. On the contrary, the dimension of the second status space Dim(S) is usually greater than the dimension of the first status space Dim(X).
Preferred developments of the invention INSERT CLAIMS 2-26.
In one preferred development, the transformation computer elements are grouped in a first hidden layer and a second hidden layer, whereby at least a part of the transformation computer elements of the first hidden layer and of the transformation computer elements of the second hidden layer are connected to one another.
At least a part of the computer elements are preferably artificial neurons.
The transformation computer elements can be connected to one another such that transformed signals can be taken at the transformation computer elements, whereby at least four transformed signals respectively relate to successive points in time.
In general, an arbitrary plurality of successive points in time can be affected by the transformed signals.
It is provided in a further development that the input computer elements are allocated to an input layer, such that the input layer comprises a plurality of sub-input layers, whereby each sub-input layer can be supplied with at least one input signal that describes a system status at respectively one point in time.
The transformation computer elements of the first hidden layer can be grouped in hidden sub-layers such that the transformation computer elements of each hidden sub-layer are respectively connected to input computer elements of a respective sub-input layer.
The connections between the computer elements can be fashioned variably and weighted.
The connections between the composite computer elements and the second output computer elements are invariant.
In a preferred embodiment, at least parts of the connections exhibit the same weighting values.
The condition can be a predetermined smoothness of the dynamic process in the space or can be a Lyapunov zero condition.
The dynamic process can be a dynamic process in a reactor, particularly in a chemical reactor, or can also be a dynamic process for modeling a traffic system; in general, any dynamic process that sequences in a technical dynamic system. Further, the arrangement or, respectively, the method can be utilized within the framework of a modeling of a financial market. In general, the methods and the arrangements are very well-suited for predicting macro-economic dynamics.
The time row values can be determined from physical signals.